For any integer \(T\) , we use \([T]\)  to denote the index set \(\lbrace 1, 2, \ldots , T\rbrace \) .
We use \(x\)  (without subscript) as shorthand for the entire vector of decisions \((x_1, \ldots , x_T) \in \mathbb {R}^n\) , and \(X\)  as the corresponding shorthand for the feasible set \(X_1 \times \ldots \times X_T\) , where we define \(n \sum _{t = 1}^T n_t\) .
For an arbitrary vector \(w = (w_1, \ldots , w_T) \in \mathbb {R}^{n}\) , we define
\(Aw \sum _{t=1}^T A_t w_t\) ,
and for any \(t \in [T]\) , we define
\(A_{\ne t}w_{\ne t} \sum _{s \in [T]\setminus \lbrace t\rbrace } A_{s} w_{s}\) .
We let \(D \mathrm {diag}(A_1, \ldots , A_T) \in \mathbb {R}^{Tm \times n}\)  denote the block-diagonal matrix with \(A_1\) , \(\ldots \) , \(A_T\)  along its diagonal.
Iterates at the \(k^\text{th}\)  iteration are denoted with superscript \(k\) .
The general normal cone [1]} to a set \(X\)  at a point \(x \in X\)  is denoted as \(N_{X}(x)\) .
For a vector \(z\)  and matrix \(M\) , we use \(\left\Vert z \right\Vert \)  and \(\left\Vert M \right\Vert \)  to denote their Euclidean and spectral norms, respectively.
When \(M\)  is positive (semi-)definite, we use \(\left\Vert z \right\Vert _M\)  to denote the (semi-)norm \(\sqrt{z^\top M z}\) .
For a vector \(z \in \mathbb {R}^N\)  and set \(S \subseteq \mathbb {R}^N\) , we define \(\mathop {\mathrm {dist}}\left(z, S\right) \min _{w \in S} \left\Vert z - w \right\Vert \) .
For any \(\epsilon > 0\) , we let \(B_\epsilon (z)\)  denote the open Euclidean ball in \(\mathbb {R}^N\)  with center \(z\)  and radius \(\epsilon \) .
We use \(\mathrm {I}\)  and \(\mathrm {0}\)  to denote the identity and zero matrices, respectively; unless indicated otherwise, their dimensions should be clear from the context.
For two square symmetric matrices \(M_1, M_2\) , we use \(M_1 \succ M_2\)  (\(M_1 \succeq M_2\) ) or \(M_2 \prec M_1\)  (\(M_2 \preceq M_2\) ) to indicate that \(M_1 - M_2\)  is positive definite (positive semidefinite).
